> Does a public police force disincentivize people from hiring private security body-guards?

Yes of course, but you are asking if you need private security for everyone using vouchers. We as a society have decided that having police is better for society instead of private security vouchers due to various reasons.

However, this question for schools remain open.

(Also, did I say the funding type of the schools matter? That’s a different discussion.)

The question remains open because different schools provide different types and quality of education (good schools, bad schools, religious schools, schools focused on science or arts, schools with different teaching methods, schools with gang violence, etc). Therefore, parents should be able to choose the schools they send their kids to despite their circumstances, and we as a society should enable them that choice.

My statement was about incentives for parents. If you want to enable parents the choices, one way to improve the incentives is school vouchers.

> Therefore, parents should be able to choose the schools they send their kids to despite their circumstances, and we as a society should enable them that choice.

I don't think you clearly explained why they can't already choose.

People vote with their wallets and their feet all the time.

At what price should "we as a society" enable parents to send their kids to any school they want ?

They can, if they have the money, which not everyone does. More people can, if they are given school vouchers. The price is less corresponding funding for public schools per student that chooses the school voucher.

So we are looking at more parents being able to choose schools vs less corresponding funding for public schools. I think it’s worth it to go for vouchers, two advantages being it gives parents choice, and it makes schools compete in a market.